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Difference between revisions of "1990 AHSME Problems/Problem 29"

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== Solution 1 ==
 
== Solution 1 ==
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Notice that inclusion of the integers between <math>34</math> to <math>100</math> inclusive is allowed as long as no integer between <math>11</math> and <math>33</math> inclusive is within the set. This provides a total of <math>100 - 34 + 1</math> = 67 solutions.  
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Notice that inclusion of the integers from <math>34</math> to <math>100</math> is allowed as long as no integer between <math>11</math> and <math>33</math> inclusive is within the set. This provides a total of <math>100 - 34 + 1</math> = 67 solutions.  
  
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Further analyzing the remaining integers between <math>1</math> and <math>10</math> , we notice that we can include all the numbers except <math>3</math> (as including <math>3</math> would force us to remove both <math>9</math> and <math>1</math>) to obtain the maximum number of  <math>9</math> solutions.  
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Further analyzation of the remaining integers between <math>1</math> and <math>10</math>, we notice that we can include all the numbers except <math>3</math> (as including <math>3</math> would force us to remove both <math>9</math> and <math>1</math>) to obtain the maximum number of  <math>9</math> solutions.  
  
 
Thus, <math>67 + 9 = 76</math>, yielding our answer, <math>\fbox{D}</math>
 
Thus, <math>67 + 9 = 76</math>, yielding our answer, <math>\fbox{D}</math>
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There are <math>\tfrac13(99-12)=29</math> multiples of three in that range, so there are <math>88-29=59</math> non-multiples, and <math>3+14+59=76</math>, which is <math>\fbox{D}</math>
 
There are <math>\tfrac13(99-12)=29</math> multiples of three in that range, so there are <math>88-29=59</math> non-multiples, and <math>3+14+59=76</math>, which is <math>\fbox{D}</math>
 
== See also ==
 
== See also ==
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{{AHSME box|year=1990|num-b=28|num-a=29}}   
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{{AHSME box|year=1990|num-b=28|num-a=30}}   
  
 
[[Category: Intermediate Number Theory Problems]]
 
[[Category: Intermediate Number Theory Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:25, 25 September 2020

Problem

A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

$\text{(A) } 50\quad \text{(B) } 66\quad \text{(C) } 67\quad \text{(D) } 76\quad \text{(E) } 78$

Solution 1

Notice that inclusion of the integers from $34$ to $100$ is allowed as long as no integer between $11$ and $33$ inclusive is within the set. This provides a total of $100 - 34 + 1$ = 67 solutions.

Further analyzation of the remaining integers between $1$ and $10$, we notice that we can include all the numbers except $3$ (as including $3$ would force us to remove both $9$ and $1$) to obtain the maximum number of $9$ solutions.

Thus, $67 + 9 = 76$, yielding our answer, $\fbox{D}$

Solution 2

Write down in a column the elements $x$ which are indivisible by three, and then follow each one by $3x, 9x, 27x, \ldots$

\[\begin{array}{ccccc}1&3&9&27&81\\2&6&18&54\\4&12&36\\5&15&45\\7&21&63\\8&24&72\\10&30&90\\11&33&99\\13&39\\\vdots&\vdots\end{array}\] We can take at most $3$ elements from the first row, and at most $2$ elements from each of the next seven rows. After that we can take only $1$ from any following row. Thus the answer is $3+7\cdot 2\,+$ the number of integers between $13$ and $100$ inclusive which are indivisible by three.

There are $\tfrac13(99-12)=29$ multiples of three in that range, so there are $88-29=59$ non-multiples, and $3+14+59=76$, which is $\fbox{D}$

See also

1990 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 28 		Followed by
Problem 30
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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